We describe the hCOSMOS redshift survey of the COSMOS field conducted with the Hectospec spectrograph on the MMT. In the central 1~deg$^2$, the hCOS20.6 subset of the survey is $>90%$ complete to a limiting $r=20.6$. The hCOSMOS survey includes 1701 new redshifts in the COSMOS field. We also use the total of 4362 new and remeasured objects to derive the age sensitive D$_n4000$ index over the entire redshift interval $0.001lesssim zlesssim0.6$. For $85%$ of the quiescent galaxies in hCOS20.6, we measure the central line-of-sight velocity dispersion. To explore potential uses of this survey, we combine previously measured galaxy sizes, profiles and stellar masses with the spectroscopy. The comparison reveals the known relations among structural, kinematic, and stellar population properties. We also compare redshift and D$_n4000$ distributions of hCOS20.6 galaxies with SHELS; a complete spectroscopic survey of 4~deg$^2$ observed to the same depth. The redshift distributions in the two fields are very different but the D$_n4000$ distribution is remarkably similar. The relation between velocity dispersion and stellar mass for massive hCOS20.6 galaxies is consistent with the local relation from SDSS. Using measured velocity dispersions, we test a photometric proxy calibrated to galaxies in the local universe. The systematic differences between the measured and photometric proxy velocity dispersions are correlated with galaxy dynamical and stellar population properties highlighting the importance of direct spectroscopic measurements.
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